LMFDB - Embedded newform 2816.2.c.j.1409.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2816,2,Mod(1409,2816)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2816.1409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2816 = 2^{8} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2816.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.4858732092$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 11)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1409.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2816.1409
Dual form			2816.2.c.j.1409.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} +2.00000 q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} +2.00000 q^{7} +2.00000 q^{9} -1.00000i q^{11} +4.00000i q^{13} -1.00000 q^{15} -2.00000 q^{17} -2.00000i q^{21} +1.00000 q^{23} +4.00000 q^{25} -5.00000i q^{27} +7.00000 q^{31} -1.00000 q^{33} -2.00000i q^{35} -3.00000i q^{37} +4.00000 q^{39} +8.00000 q^{41} +6.00000i q^{43} -2.00000i q^{45} +8.00000 q^{47} -3.00000 q^{49} +2.00000i q^{51} +6.00000i q^{53} -1.00000 q^{55} -5.00000i q^{59} +12.0000i q^{61} +4.00000 q^{63} +4.00000 q^{65} -7.00000i q^{67} -1.00000i q^{69} +3.00000 q^{71} -4.00000 q^{73} -4.00000i q^{75} -2.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +2.00000i q^{85} -15.0000 q^{89} +8.00000i q^{91} -7.00000i q^{93} -7.00000 q^{97} -2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^7 + 4 * q^9	$ 2 q + 4 q^{7} + 4 q^{9} - 2 q^{15} - 4 q^{17} + 2 q^{23} + 8 q^{25} + 14 q^{31} - 2 q^{33} + 8 q^{39} + 16 q^{41} + 16 q^{47} - 6 q^{49} - 2 q^{55} + 8 q^{63} + 8 q^{65} + 6 q^{71} - 8 q^{73} - 20 q^{79} + 2 q^{81} - 30 q^{89} - 14 q^{97}+O(q^{100}) $ 2 * q + 4 * q^7 + 4 * q^9 - 2 * q^15 - 4 * q^17 + 2 * q^23 + 8 * q^25 + 14 * q^31 - 2 * q^33 + 8 * q^39 + 16 * q^41 + 16 * q^47 - 6 * q^49 - 2 * q^55 + 8 * q^63 + 8 * q^65 + 6 * q^71 - 8 * q^73 - 20 * q^79 + 2 * q^81 - 30 * q^89 - 14 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2816\mathbb{Z}\right)^\times$.

$n$	$1025$	$1541$	$2047$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i	−0.974679	−	0.223607i	$-0.928217\pi$
$5$	− 1.00000i	− 0.447214i	0.974679	−	0.223607i	$-0.0717831\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	− 2.00000i	− 0.298142i
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	2.00000i	0.280056i
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5.00000i	− 0.650945i	−0.945552	−	0.325472i	$-0.894477\pi$
$59$	− 5.00000i	− 0.650945i	0.945552	−	0.325472i	$-0.105523\pi$
$60$	0	0
$61$	12.0000i	1.53644i	0.640184	+	0.768221i	$0.278858\pi$
$61$	12.0000i	1.53644i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	0	0
$69$	− 1.00000i	− 0.120386i
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	− 4.00000i	− 0.461880i
$76$	0	0
$77$	− 2.00000i	− 0.227921i
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	2.00000i	0.216930i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	8.00000i	0.838628i
$92$	0	0
$93$	− 7.00000i	− 0.725866i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	0	0
$99$	− 2.00000i	− 0.201008i
$100$	0	0
$101$	− 2.00000i	− 0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$	− 2.00000i	− 0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	0	0
$109$	10.0000i	0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$	10.0000i	0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	− 1.00000i	− 0.0932505i
$116$	0	0
$117$	8.00000i	0.739600i
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 8.00000i	− 0.721336i
$124$	0	0
$125$	− 9.00000i	− 0.804984i
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	− 18.0000i	− 1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$	− 18.0000i	− 1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	7.00000	0.598050	0.299025	−	0.954245i	$-0.403339\pi$
$137$	7.00000	0.598050	0.299025	+	0.954245i	$0.403339\pi$
$138$	0	0
$139$	− 10.0000i	− 0.848189i	−0.905618	−	0.424094i	$-0.860592\pi$
$139$	− 10.0000i	− 0.848189i	0.905618	−	0.424094i	$-0.139408\pi$
$140$	0	0
$141$	− 8.00000i	− 0.673722i
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000i	0.247436i
$148$	0	0
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	− 7.00000i	− 0.562254i
$156$	0	0
$157$	− 7.00000i	− 0.558661i	−0.960195	−	0.279330i	$-0.909888\pi$
$157$	− 7.00000i	− 0.558661i	0.960195	−	0.279330i	$-0.0901125\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	1.00000i	0.0778499i
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	0	0
$177$	−5.00000	−0.375823
$178$	0	0
$179$	− 15.0000i	− 1.12115i	−0.828103	−	0.560576i	$-0.810580\pi$
$179$	− 15.0000i	− 1.12115i	0.828103	−	0.560576i	$-0.189420\pi$
$180$	0	0
$181$	− 7.00000i	− 0.520306i	−0.965567	−	0.260153i	$-0.916227\pi$
$181$	− 7.00000i	− 0.520306i	0.965567	−	0.260153i	$-0.0837730\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	0	0
$185$	−3.00000	−0.220564
$186$	0	0
$187$	2.00000i	0.146254i
$188$	0	0
$189$	− 10.0000i	− 0.727393i
$190$	0	0
$191$	17.0000	1.23008	0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000	1.23008	0.615038	+	0.788497i	$0.289140\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	− 4.00000i	− 0.286446i
$196$	0	0
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 8.00000i	− 0.558744i
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.0000i	0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$	12.0000i	0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	0	0
$213$	− 3.00000i	− 0.205557i
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	14.0000	0.950382
$218$	0	0
$219$	4.00000i	0.270295i
$220$	0	0
$221$	− 8.00000i	− 0.538138i
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	− 15.0000i	− 0.991228i	−0.868543	−	0.495614i	$-0.834943\pi$
$229$	− 15.0000i	− 0.991228i	0.868543	−	0.495614i	$-0.165057\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	− 8.00000i	− 0.521862i
$236$	0	0
$237$	10.0000i	0.649570i
$238$	0	0
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	3.00000i	0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	23.0000i	1.45175i	0.687828	+	0.725874i	$0.258564\pi$
$251$	23.0000i	1.45175i	−0.687828	+	0.725874i	$0.741436\pi$
$252$	0	0
$253$	− 1.00000i	− 0.0628695i
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	− 6.00000i	− 0.372822i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	15.0000i	0.917985i
$268$	0	0
$269$	10.0000i	0.609711i	0.952399	+	0.304855i	$0.0986081\pi$
$269$	10.0000i	0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	0	0
$279$	14.0000	0.838158
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.0000	0.944450
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	7.00000i	0.410347i
$292$	0	0
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	0	0
$295$	−5.00000	−0.291111
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	4.00000i	0.231326i
$300$	0	0
$301$	12.0000i	0.691669i
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	0	0
$309$	− 16.0000i	− 0.910208i
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	− 4.00000i	− 0.225374i
$316$	0	0
$317$	13.0000i	0.730153i	0.930978	+	0.365076i	$0.118957\pi$
$317$	13.0000i	0.730153i	−0.930978	+	0.365076i	$0.881043\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	0	0
$324$	0	0
$325$	16.0000i	0.887520i
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	− 7.00000i	− 0.384755i	−0.981321	−	0.192377i	$-0.938380\pi$
$331$	− 7.00000i	− 0.384755i	0.981321	−	0.192377i	$-0.0616198\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	−7.00000	−0.382451
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	− 9.00000i	− 0.488813i
$340$	0	0
$341$	− 7.00000i	− 0.379071i
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	− 28.0000i	− 1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$	− 28.0000i	− 1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$	0	0
$349$	30.0000i	1.60586i	0.596071	+	0.802932i	$0.296728\pi$
$349$	30.0000i	1.60586i	−0.596071	+	0.802932i	$0.703272\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	0	0
$353$	−21.0000	−1.11772	−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000	−1.11772	−0.558859	+	0.829263i	$0.688761\pi$
$354$	0	0
$355$	− 3.00000i	− 0.159223i
$356$	0	0
$357$	4.00000i	0.211702i
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	1.00000i	0.0524864i
$364$	0	0
$365$	4.00000i	0.209370i
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	0	0
$369$	16.0000	0.832927
$370$	0	0
$371$	12.0000i	0.623009i
$372$	0	0
$373$	26.0000i	1.34623i	0.739538	+	0.673114i	$0.235044\pi$
$373$	26.0000i	1.34623i	−0.739538	+	0.673114i	$0.764956\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	0	0
$378$	0	0
$379$	5.00000i	0.256833i	0.991720	+	0.128416i	$0.0409894\pi$
$379$	5.00000i	0.256833i	−0.991720	+	0.128416i	$0.959011\pi$
$380$	0	0
$381$	− 8.00000i	− 0.409852i
$382$	0	0
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	12.0000i	0.609994i
$388$	0	0
$389$	15.0000i	0.760530i	0.924878	+	0.380265i	$0.124167\pi$
$389$	15.0000i	0.760530i	−0.924878	+	0.380265i	$0.875833\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	−18.0000	−0.907980
$394$	0	0
$395$	10.0000i	0.503155i
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	28.0000i	1.39478i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	−3.00000	−0.148704
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	− 7.00000i	− 0.345285i
$412$	0	0
$413$	− 10.0000i	− 0.492068i
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−10.0000	−0.489702
$418$	0	0
$419$	20.0000i	0.977064i	0.872546	+	0.488532i	$0.162467\pi$
$419$	20.0000i	0.977064i	−0.872546	+	0.488532i	$0.837533\pi$
$420$	0	0
$421$	− 22.0000i	− 1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	− 22.0000i	− 1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$	0	0
$423$	16.0000	0.777947
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	24.0000i	1.16144i
$428$	0	0
$429$	− 4.00000i	− 0.193122i
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	11.0000i	0.522626i	0.965254	+	0.261313i	$0.0841554\pi$
$443$	11.0000i	0.522626i	−0.965254	+	0.261313i	$0.915845\pi$
$444$	0	0
$445$	15.0000i	0.711068i
$446$	0	0
$447$	10.0000	0.472984
$448$	0	0
$449$	35.0000	1.65175	0.825876	−	0.563852i	$-0.190681\pi$
$449$	35.0000	1.65175	0.825876	+	0.563852i	$0.190681\pi$
$450$	0	0
$451$	− 8.00000i	− 0.376705i
$452$	0	0
$453$	2.00000i	0.0939682i
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	0	0
$459$	10.0000i	0.466760i
$460$	0	0
$461$	12.0000i	0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$	12.0000i	0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	0	0
$463$	−11.0000	−0.511213	−0.255607	−	0.966781i	$-0.582275\pi$
$463$	−11.0000	−0.511213	−0.255607	+	0.966781i	$0.582275\pi$
$464$	0	0
$465$	−7.00000	−0.324617
$466$	0	0
$467$	− 27.0000i	− 1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$	− 27.0000i	− 1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$	0	0
$469$	− 14.0000i	− 0.646460i
$470$	0	0
$471$	−7.00000	−0.322543
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	− 2.00000i	− 0.0910032i
$484$	0	0
$485$	7.00000i	0.317854i
$486$	0	0
$487$	−23.0000	−1.04223	−0.521115	−	0.853487i	$-0.674484\pi$
$487$	−23.0000	−1.04223	−0.521115	+	0.853487i	$0.674484\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	8.00000i	0.361035i	0.983572	+	0.180517i	$0.0577772\pi$
$491$	8.00000i	0.361035i	−0.983572	+	0.180517i	$0.942223\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	20.0000i	0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$	20.0000i	0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$	0	0
$501$	− 12.0000i	− 0.536120i
$502$	0	0
$503$	26.0000	1.15928	0.579641	−	0.814872i	$-0.303193\pi$
$503$	26.0000	1.15928	0.579641	+	0.814872i	$0.303193\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	15.0000i	0.664863i	0.943127	+	0.332432i	$0.107869\pi$
$509$	15.0000i	0.664863i	−0.943127	+	0.332432i	$0.892131\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 16.0000i	− 0.705044i
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	0	0
$525$	− 8.00000i	− 0.349149i
$526$	0	0
$527$	−14.0000	−0.609850
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	− 10.0000i	− 0.433963i
$532$	0	0
$533$	32.0000i	1.38607i
$534$	0	0
$535$	−18.0000	−0.778208
$536$	0	0
$537$	−15.0000	−0.647298
$538$	0	0
$539$	3.00000i	0.129219i
$540$	0	0
$541$	− 8.00000i	− 0.343947i	−0.985102	−	0.171973i	$-0.944986\pi$
$541$	− 8.00000i	− 0.343947i	0.985102	−	0.171973i	$-0.0550143\pi$
$542$	0	0
$543$	−7.00000	−0.300399
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	24.0000i	1.02430i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	3.00000i	0.127343i
$556$	0	0
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	− 9.00000i	− 0.378633i
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	28.0000i	1.17176i	0.810397	+	0.585882i	$0.199252\pi$
$571$	28.0000i	1.17176i	−0.810397	+	0.585882i	$0.800748\pi$
$572$	0	0
$573$	− 17.0000i	− 0.710185i
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	33.0000	1.37381	0.686904	−	0.726748i	$-0.258969\pi$
$577$	33.0000	1.37381	0.686904	+	0.726748i	$0.258969\pi$
$578$	0	0
$579$	− 4.00000i	− 0.166234i
$580$	0	0
$581$	− 12.0000i	− 0.497844i
$582$	0	0
$583$	6.00000	0.248495
$584$	0	0
$585$	8.00000	0.330759
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	44.0000	1.80686	0.903432	−	0.428732i	$-0.141040\pi$
$593$	44.0000	1.80686	0.903432	+	0.428732i	$0.141040\pi$
$594$	0	0
$595$	4.00000i	0.163984i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	− 14.0000i	− 0.570124i
$604$	0	0
$605$	1.00000i	0.0406558i
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	32.0000i	1.29458i
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	−8.00000	−0.322591
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	25.0000i	1.00483i	0.864625	+	0.502417i	$0.167556\pi$
$619$	25.0000i	1.00483i	−0.864625	+	0.502417i	$0.832444\pi$
$620$	0	0
$621$	− 5.00000i	− 0.200643i
$622$	0	0
$623$	−30.0000	−1.20192
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.00000i	0.239236i
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	− 8.00000i	− 0.317470i
$636$	0	0
$637$	− 12.0000i	− 0.475457i
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	29.0000i	1.14365i	0.820376	+	0.571824i	$0.193764\pi$
$643$	29.0000i	1.14365i	−0.820376	+	0.571824i	$0.806236\pi$
$644$	0	0
$645$	− 6.00000i	− 0.236250i
$646$	0	0
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	0	0
$649$	−5.00000	−0.196267
$650$	0	0
$651$	− 14.0000i	− 0.548703i
$652$	0	0
$653$	− 41.0000i	− 1.60445i	−0.597019	−	0.802227i	$-0.703648\pi$
$653$	− 41.0000i	− 1.60445i	0.597019	−	0.802227i	$-0.296352\pi$
$654$	0	0
$655$	−18.0000	−0.703318
$656$	0	0
$657$	−8.00000	−0.312110
$658$	0	0
$659$	10.0000i	0.389545i	0.980848	+	0.194772i	$0.0623968\pi$
$659$	10.0000i	0.389545i	−0.980848	+	0.194772i	$0.937603\pi$
$660$	0	0
$661$	− 37.0000i	− 1.43913i	−0.694423	−	0.719567i	$-0.744340\pi$
$661$	− 37.0000i	− 1.43913i	0.694423	−	0.719567i	$-0.255660\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	− 19.0000i	− 0.734582i
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	− 20.0000i	− 0.769800i
$676$	0	0
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	0	0
$685$	− 7.00000i	− 0.267456i
$686$	0	0
$687$	−15.0000	−0.572286
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	17.0000i	0.646710i	0.946278	+	0.323355i	$0.104811\pi$
$691$	17.0000i	0.646710i	−0.946278	+	0.323355i	$0.895189\pi$
$692$	0	0
$693$	− 4.00000i	− 0.151947i
$694$	0	0
$695$	−10.0000	−0.379322
$696$	0	0
$697$	−16.0000	−0.606043
$698$	0	0
$699$	24.0000i	0.907763i
$700$	0	0
$701$	2.00000i	0.0755390i	0.999286	+	0.0377695i	$0.0120253\pi$
$701$	2.00000i	0.0755390i	−0.999286	+	0.0377695i	$0.987975\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	25.0000i	0.938895i	0.882960	+	0.469447i	$0.155547\pi$
$709$	25.0000i	0.938895i	−0.882960	+	0.469447i	$0.844453\pi$
$710$	0	0
$711$	−20.0000	−0.750059
$712$	0	0
$713$	7.00000	0.262152
$714$	0	0
$715$	− 4.00000i	− 0.149592i
$716$	0	0
$717$	30.0000i	1.12037i
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	8.00000i	0.297523i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.00000	−0.111264	−0.0556319	−	0.998451i	$-0.517717\pi$
$727$	−3.00000	−0.111264	−0.0556319	+	0.998451i	$0.517717\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 12.0000i	− 0.443836i
$732$	0	0
$733$	− 36.0000i	− 1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$	− 36.0000i	− 1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	−7.00000	−0.257848
$738$	0	0
$739$	50.0000i	1.83928i	0.392763	+	0.919640i	$0.371519\pi$
$739$	50.0000i	1.83928i	−0.392763	+	0.919640i	$0.628481\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	− 12.0000i	− 0.439057i
$748$	0	0
$749$	− 36.0000i	− 1.31541i
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	0	0
$753$	23.0000	0.838167
$754$	0	0
$755$	2.00000i	0.0727875i
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	0	0
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	20.0000i	0.724049i
$764$	0	0
$765$	4.00000i	0.144620i
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	2.00000i	0.0720282i
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	28.0000	1.00579
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 3.00000i	− 0.107348i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.00000	−0.249841
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	0	0
$789$	14.0000i	0.498413i
$790$	0	0
$791$	18.0000	0.640006
$792$	0	0
$793$	−48.0000	−1.70453
$794$	0	0
$795$	− 6.00000i	− 0.212798i
$796$	0	0
$797$	53.0000i	1.87736i	0.344795	+	0.938678i	$0.387949\pi$
$797$	53.0000i	1.87736i	−0.344795	+	0.938678i	$0.612051\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	− 2.00000i	− 0.0704907i
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	38.0000i	1.33436i	0.744896	+	0.667180i	$0.232499\pi$
$811$	38.0000i	1.33436i	−0.744896	+	0.667180i	$0.767501\pi$
$812$	0	0
$813$	28.0000i	0.982003i
$814$	0	0
$815$	4.00000	0.140114
$816$	0	0
$817$	0	0
$818$	0	0
$819$	16.0000i	0.559085i
$820$	0	0
$821$	− 22.0000i	− 0.767805i	−0.923374	−	0.383903i	$-0.874580\pi$
$821$	− 22.0000i	− 0.767805i	0.923374	−	0.383903i	$-0.125420\pi$
$822$	0	0
$823$	−39.0000	−1.35945	−0.679727	−	0.733465i	$-0.737902\pi$
$823$	−39.0000	−1.35945	−0.679727	+	0.733465i	$0.737902\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	52.0000i	1.80822i	0.427303	+	0.904109i	$0.359464\pi$
$827$	52.0000i	1.80822i	−0.427303	+	0.904109i	$0.640536\pi$
$828$	0	0
$829$	25.0000i	0.868286i	0.900844	+	0.434143i	$0.142949\pi$
$829$	25.0000i	0.868286i	−0.900844	+	0.434143i	$0.857051\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	− 12.0000i	− 0.415277i
$836$	0	0
$837$	− 35.0000i	− 1.20978i
$838$	0	0
$839$	5.00000	0.172619	0.0863096	−	0.996268i	$-0.472493\pi$
$839$	5.00000	0.172619	0.0863096	+	0.996268i	$0.472493\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	3.00000i	0.103203i
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 3.00000i	− 0.102839i
$852$	0	0
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−8.00000	−0.273275	−0.136637	−	0.990621i	$-0.543630\pi$
$857$	−8.00000	−0.273275	−0.136637	+	0.990621i	$0.543630\pi$
$858$	0	0
$859$	15.0000i	0.511793i	0.966704	+	0.255897i	$0.0823707\pi$
$859$	15.0000i	0.511793i	−0.966704	+	0.255897i	$0.917629\pi$
$860$	0	0
$861$	− 16.0000i	− 0.545279i
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	13.0000i	0.441503i
$868$	0	0
$869$	10.0000i	0.339227i
$870$	0	0
$871$	28.0000	0.948744
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	− 18.0000i	− 0.608511i
$876$	0	0
$877$	− 12.0000i	− 0.405211i	−0.979260	−	0.202606i	$-0.935059\pi$
$877$	− 12.0000i	− 0.405211i	0.979260	−	0.202606i	$-0.0649409\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−43.0000	−1.44871	−0.724353	−	0.689429i	$-0.757862\pi$
$881$	−43.0000	−1.44871	−0.724353	+	0.689429i	$0.757862\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	5.00000i	0.168073i
$886$	0	0
$887$	22.0000	0.738688	0.369344	−	0.929293i	$-0.379582\pi$
$887$	22.0000	0.738688	0.369344	+	0.929293i	$0.379582\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	− 1.00000i	− 0.0335013i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−15.0000	−0.501395
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	−7.00000	−0.232688
$906$	0	0
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	0	0
$909$	− 4.00000i	− 0.132672i
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	0	0
$915$	− 12.0000i	− 0.396708i
$916$	0	0
$917$	− 36.0000i	− 1.18882i
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	0	0
$923$	12.0000i	0.394985i
$924$	0	0
$925$	− 12.0000i	− 0.394558i
$926$	0	0
$927$	32.0000	1.05102
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	12.0000i	0.392862i
$934$	0	0
$935$	2.00000	0.0654070
$936$	0	0
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	0	0
$939$	− 1.00000i	− 0.0326338i
$940$	0	0
$941$	42.0000i	1.36916i	0.728937	+	0.684580i	$0.240015\pi$
$941$	42.0000i	1.36916i	−0.728937	+	0.684580i	$0.759985\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	−10.0000	−0.325300
$946$	0	0
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	− 16.0000i	− 0.519382i
$950$	0	0
$951$	13.0000	0.421554
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	0	0
$955$	− 17.0000i	− 0.550107i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.0000	0.452084
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	− 36.0000i	− 1.16008i
$964$	0	0
$965$	− 4.00000i	− 0.128765i
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 47.0000i	− 1.50830i	−0.656701	−	0.754151i	$-0.728049\pi$
$971$	− 47.0000i	− 1.50830i	0.656701	−	0.754151i	$-0.271951\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	0	0
$975$	16.0000	0.512410
$976$	0	0
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	0	0
$979$	15.0000i	0.479402i
$980$	0	0
$981$	20.0000i	0.638551i
$982$	0	0
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	6.00000i	0.190789i
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	−7.00000	−0.222138
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 38.0000i	− 1.20347i	−0.798695	−	0.601736i	$-0.794476\pi$
$997$	− 38.0000i	− 1.20347i	0.798695	−	0.601736i	$-0.205524\pi$
$998$	0	0
$999$	−15.0000	−0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2816.2.c.j.1409.1		2
4.3	odd	2		2816.2.c.f.1409.2		2
8.3	odd	2		2816.2.c.f.1409.1		2
8.5	even	2	inner	2816.2.c.j.1409.2		2
16.3	odd	4		704.2.a.c.1.1		1
16.5	even	4		11.2.a.a.1.1	✓	1
16.11	odd	4		176.2.a.b.1.1		1
16.13	even	4		704.2.a.h.1.1		1
48.5	odd	4		99.2.a.d.1.1		1
48.11	even	4		1584.2.a.g.1.1		1
48.29	odd	4		6336.2.a.br.1.1		1
48.35	even	4		6336.2.a.bu.1.1		1
80.27	even	4		4400.2.b.h.4049.1		2
80.37	odd	4		275.2.b.a.199.1		2
80.43	even	4		4400.2.b.h.4049.2		2
80.53	odd	4		275.2.b.a.199.2		2
80.59	odd	4		4400.2.a.i.1.1		1
80.69	even	4		275.2.a.b.1.1		1
112.5	odd	12		539.2.e.g.67.1		2
112.27	even	4		8624.2.a.j.1.1		1
112.37	even	12		539.2.e.h.67.1		2
112.53	even	12		539.2.e.h.177.1		2
112.69	odd	4		539.2.a.a.1.1		1
112.101	odd	12		539.2.e.g.177.1		2
144.5	odd	12		891.2.e.b.298.1		2
144.85	even	12		891.2.e.k.298.1		2
144.101	odd	12		891.2.e.b.595.1		2
144.133	even	12		891.2.e.k.595.1		2
176.5	even	20		121.2.c.e.3.1		4
176.21	odd	4		121.2.a.d.1.1		1
176.37	even	20		121.2.c.e.27.1		4
176.43	even	4		1936.2.a.i.1.1		1
176.53	even	20		121.2.c.e.81.1		4
176.69	even	20		121.2.c.e.9.1		4
176.85	odd	20		121.2.c.a.9.1		4
176.101	odd	20		121.2.c.a.81.1		4
176.109	odd	4		7744.2.a.x.1.1		1
176.117	odd	20		121.2.c.a.27.1		4
176.131	even	4		7744.2.a.k.1.1		1
176.149	odd	20		121.2.c.a.3.1		4
208.181	even	4		1859.2.a.b.1.1		1
240.53	even	4		2475.2.c.a.199.1		2
240.149	odd	4		2475.2.a.a.1.1		1
240.197	even	4		2475.2.c.a.199.2		2
272.101	even	4		3179.2.a.a.1.1		1
304.37	odd	4		3971.2.a.b.1.1		1
336.293	even	4		4851.2.a.t.1.1		1
368.229	odd	4		5819.2.a.a.1.1		1
464.405	even	4		9251.2.a.d.1.1		1
528.197	even	4		1089.2.a.b.1.1		1
880.549	odd	4		3025.2.a.a.1.1		1
1232.1077	even	4		5929.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.2.a.a.1.1	✓	1	16.5	even	4
99.2.a.d.1.1		1	48.5	odd	4
121.2.a.d.1.1		1	176.21	odd	4
121.2.c.a.3.1		4	176.149	odd	20
121.2.c.a.9.1		4	176.85	odd	20
121.2.c.a.27.1		4	176.117	odd	20
121.2.c.a.81.1		4	176.101	odd	20
121.2.c.e.3.1		4	176.5	even	20
121.2.c.e.9.1		4	176.69	even	20
121.2.c.e.27.1		4	176.37	even	20
121.2.c.e.81.1		4	176.53	even	20
176.2.a.b.1.1		1	16.11	odd	4
275.2.a.b.1.1		1	80.69	even	4
275.2.b.a.199.1		2	80.37	odd	4
275.2.b.a.199.2		2	80.53	odd	4
539.2.a.a.1.1		1	112.69	odd	4
539.2.e.g.67.1		2	112.5	odd	12
539.2.e.g.177.1		2	112.101	odd	12
539.2.e.h.67.1		2	112.37	even	12
539.2.e.h.177.1		2	112.53	even	12
704.2.a.c.1.1		1	16.3	odd	4
704.2.a.h.1.1		1	16.13	even	4
891.2.e.b.298.1		2	144.5	odd	12
891.2.e.b.595.1		2	144.101	odd	12
891.2.e.k.298.1		2	144.85	even	12
891.2.e.k.595.1		2	144.133	even	12
1089.2.a.b.1.1		1	528.197	even	4
1584.2.a.g.1.1		1	48.11	even	4
1859.2.a.b.1.1		1	208.181	even	4
1936.2.a.i.1.1		1	176.43	even	4
2475.2.a.a.1.1		1	240.149	odd	4
2475.2.c.a.199.1		2	240.53	even	4
2475.2.c.a.199.2		2	240.197	even	4
2816.2.c.f.1409.1		2	8.3	odd	2
2816.2.c.f.1409.2		2	4.3	odd	2
2816.2.c.j.1409.1		2	1.1	even	1	trivial
2816.2.c.j.1409.2		2	8.5	even	2	inner
3025.2.a.a.1.1		1	880.549	odd	4
3179.2.a.a.1.1		1	272.101	even	4
3971.2.a.b.1.1		1	304.37	odd	4
4400.2.a.i.1.1		1	80.59	odd	4
4400.2.b.h.4049.1		2	80.27	even	4
4400.2.b.h.4049.2		2	80.43	even	4
4851.2.a.t.1.1		1	336.293	even	4
5819.2.a.a.1.1		1	368.229	odd	4
5929.2.a.h.1.1		1	1232.1077	even	4
6336.2.a.br.1.1		1	48.29	odd	4
6336.2.a.bu.1.1		1	48.35	even	4
7744.2.a.k.1.1		1	176.131	even	4
7744.2.a.x.1.1		1	176.109	odd	4
8624.2.a.j.1.1		1	112.27	even	4
9251.2.a.d.1.1		1	464.405	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2816 = 2^{8} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2816.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} +2.00000 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} +2.00000 q^{7} +2.00000 q^{9} -1.00000i q^{11} +4.00000i q^{13} -1.00000 q^{15} -2.00000 q^{17} -2.00000i q^{21} +1.00000 q^{23} +4.00000 q^{25} -5.00000i q^{27} +7.00000 q^{31} -1.00000 q^{33} -2.00000i q^{35} -3.00000i q^{37} +4.00000 q^{39} +8.00000 q^{41} +6.00000i q^{43} -2.00000i q^{45} +8.00000 q^{47} -3.00000 q^{49} +2.00000i q^{51} +6.00000i q^{53} -1.00000 q^{55} -5.00000i q^{59} +12.0000i q^{61} +4.00000 q^{63} +4.00000 q^{65} -7.00000i q^{67} -1.00000i q^{69} +3.00000 q^{71} -4.00000 q^{73} -4.00000i q^{75} -2.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} +2.00000i q^{85} -15.0000 q^{89} +8.00000i q^{91} -7.00000i q^{93} -7.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^7 + 4 * q^9	\( 2 q + 4 q^{7} + 4 q^{9} - 2 q^{15} - 4 q^{17} + 2 q^{23} + 8 q^{25} + 14 q^{31} - 2 q^{33} + 8 q^{39} + 16 q^{41} + 16 q^{47} - 6 q^{49} - 2 q^{55} + 8 q^{63} + 8 q^{65} + 6 q^{71} - 8 q^{73} - 20 q^{79} + 2 q^{81} - 30 q^{89} - 14 q^{97}+O(q^{100}) \) 2 * q + 4 * q^7 + 4 * q^9 - 2 * q^15 - 4 * q^17 + 2 * q^23 + 8 * q^25 + 14 * q^31 - 2 * q^33 + 8 * q^39 + 16 * q^41 + 16 * q^47 - 6 * q^49 - 2 * q^55 + 8 * q^63 + 8 * q^65 + 6 * q^71 - 8 * q^73 - 20 * q^79 + 2 * q^81 - 30 * q^89 - 14 * q^97

Introduction

L-functions

Modular forms

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Representations

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Database

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